Let B* be the dual space of a Banach space B. Recall the definition of the duality mapping (set-valued mapping) $$ J: B\to 2^{B^*}, J(f):=\{L\in B^*: L(f)=||L||||f|, ||L||=||f||\}. $$ For instance, if $B=L^p (1<p<+\infty)$, then $$ J(f)=\frac{\overline{f}|f|^{p-1}}{||f||_{L^p}^{p-2}},\ 0\ne f\in L^p. $$ My question is about the characterization of the duality mapping of Banach space $C_0(X)$ of all continuous functions vanishing at infinity on a locally compact Hausdorff space $X$. The space $C_0(X)$ is endowed with the maximum norm $||\cdot||_{C_0(X)}$. The dual space of $C_0(X)$ is the Banach space $M(X)$ of all the complex-valued regular Borel measures on $X$ with a bounded total variation [Rudin, Real and Complex Analysis, 1987, Theorem 6.19]. In other words, every continuous functional $T$ on $C_0(X)$ is represented by a $\mu\in M(X)$ in the sense that $$ T(f)=\int_X f(x)d\mu(x),\mbox{ for all }f\in C_0(X). $$

For every $f\in C_0(X)$, there exists a maximum. The duality mapping $J(f)$ must be closely related the set of maximum points of the continuous function $f$, i.e, $$ A=\{x\in X: |f(x)|=||f||_{C_0(X)}\}. $$ Sufficiency: (i) the point evaluation functional $\delta_x\in J(f)$ for all $x\in A$; (ii) the indicator function $\frac{1}{|C|}1_C\in J(f)$ for all $C\subseteq A$, where $1_C(x):=1$ if $x\in C$ and $0$ otherwise.

Is the above sufficiency result also necessary?

My question is how to characterize $J(f)$ for any $f\in C_0(X)$.

Many thanks for your kind comments or references in advance.

Let B* be the dual space of a Banach space B. Recall the definition of the duality mapping (set-valued mapping) $$ J: B\to 2^{B^*}, J(f):=\{L\in B^*: L(f)=||L||||f|, ||L||=||f||\}. $$ For instance, if $B=L^p (1<p<+\infty)$, then $$ J(f)=\frac{\overline{f}|f|^{p-1}}{||f||_{L^p}^{p-2}},\ 0\ne f\in L^p. $$ My question is about the characterization of the duality mapping of Banach space $C_0(X)$ of all continuous functions vanishing at infinity on a locally compact Hausdorff space $X$. The space $C_0(X)$ is endowed with the maximum norm $||\cdot||_{C_0(X)}$. The dual space of $C_0(X)$ is the Banach space $M(X)$ of all the complex-valued regular Borel measures on $X$ with a bounded total variation [Rudin, Real and Complex Analysis, 1987, Theorem 6.19]. In other words, every continuous functional $T$ on $C_0(X)$ is represented by a $\mu\in M(X)$ in the sense that $$ T(f)=\int_X f(x)d\mu(x),\mbox{ for all }f\in C_0(X). $$

For every $f\in C_0(X)$, there exists a maximum. The duality mapping $J(f)$ must be closely related the set of maximum points of the continuous function $f$, i.e, $$ A=\{x\in X: |f(x)|=||f||_{C_0(X)}\}. $$

My question is how to characterize $J(f)$ for any $f\in C_0(X)$.

Many thanks for your kind comments or references in advance.